On the exceptional locus of the birational projections of normal surface singularity into a plane

Given a normal surface singularity $(X, Q)$ and a birational morphism to a non- singular surface $\pi : X \to S$, we investigate the local geometry of the exceptional divisor $L$ of $\pi$. We prove that the dimension of the tangent space to $L$ at $Q$ equals the number of exceptional components meeting at $Q$. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterization of minimal singularities is obtained in these terms.Comment: 12 pages, 2 figure

Geometry of the Kimura 3-parameter model

The Kimura 3-parameter model on a tree of n leaves is one of the most used in phylogenetics. The affine algebraic variety W associated to it is a toric variety. We study its geometry and we prove that it is isomorphic to a geometric quotient of the affine space by a finite group acting on it. As a consequence, we are able to study the singularities of W and prove that the biologically meaningful points are smooth points. Then we give an algorithm for constructing a set of minimal generators of the localized ideal at these points, for an arbitrary number of leaves n. This leads to a major improvement of phylogenetic reconstruction methods based on algebraic geometry.Comment: 26 pages with 4 figure

Equisingularity classes of birational projections of normal singularities to a plane

Given a birational normal extension S of a two-dimensional local regular ring R, we describe all the equisingularity types of the complete ideals J in R whose blowing-up has some point at which the local ring is analytically isomorphic to S. The problem of classifying the germs of such normal surface singularities was already posed by Spivakovsky (Ann. of Math. 1990). This problem has two parts: discrete and continous. The continous part is to some extent equivalent to the problem of the moduli of plane curve singularities, while the main result of this paper solves completely the discrete part.Comment: 22 pages, 3 figures. To appear in Advances in Mathematic

Evaluating Labor Market Reforms: A General Equilibrium Approach

fixed-term contracts, firing costs, general equilibrium, heterogeneous agents

The space of phylogenetic mixtures for equivariant models

The selection of the most suitable evolutionary model to analyze the given molecular data is usually left to biologist's choice. In his famous book, J Felsenstein suggested that certain linear equations satisfied by the expected probabilities of patterns observed at the leaves of a phylogenetic tree could be used for model selection. It remained open the question regarding whether these equations were enough for characterizing the evolutionary model. Here we prove that, for equivariant models of evolution, the space of distributions satisfying these linear equations coincides with the space of distributions arising from mixtures of trees on a set of taxa. In other words, we prove that an alignment is produced from a mixture of phylogenetic trees under an equivariant evolutionary model if and only if its distribution of column patterns satisfies the linear equations mentioned above. Moreover, for each equivariant model and for any number of taxa, we provide a set of linearly independent equations defining this space of phylogenetic mixtures. This is a powerful tool that has already been successfully used in model selection. We also use the results obtained to study identifiability issues for phylogenetic mixtures.Comment: 28 pages, 1 figure; to appear in Algorithms for Molecular Biolog

Detection of growth-related QTLs in turbot (Scophtalmus maximux)

Background The turbot (Scophthalmus maximus) is a highly appreciated European aquaculture species. Growth related traits constitute the main goal of the ongoing genetic breeding programs of this species. The recent construction of a consensus linkage map in this species has allowed the selection of a panel of 100 homogeneously distributed markers covering the 26 linkage groups (LG) suitable for QTL search. In this study we addressed the detection of QTL with effect on body weight, length and Fulton's condition factor. Results Eight families from two genetic breeding programs comprising 814 individuals were used to search for growth related QTL using the panel of microsatellites available for QTL screening. Two different approaches, maximum likelihood and regression interval mapping, were used in order to search for QTL. Up to eleven significant QTL were detected with both methods in at least one family: four for weight on LGs 5, 14, 15 and 16; five for length on LGs 5, 6, 12, 14 and 15; and two for Fulton's condition factor on LGs 3 and 16. In these LGs an association analysis was performed to ascertain the microsatellite marker with the highest apparent effect on the trait, in order to test the possibility of using them for marker assisted selection. Conclusions The use of regression interval mapping and maximum likelihood methods for QTL detection provided consistent results in many cases, although the high variation observed for traits mean among families made it difficult to evaluate QTL effects. Finer mapping of detected QTL, looking for tightly linked markers to the causative mutation, and comparative genomics are suggested to deepen in the analysis of QTL in turbot so they can be applied in marker assisted selection programs